# 'Find the equation of the tangent to the curve ay^2=x^3 at the point (at^2,at^3) where a>0 and t is a parameter ?

# y = 3/2tx - 1/2at^3 #

We have a family of curves defined for

# C(a) : " " ay^2 = x^3 # ..... [A]Or in parametric form:

# C(a) : " " { (x(t)=at^2), (y(t)=at^3) :} # Here we have a plot for

#a=1#

If we differentiate equation [A] implicitly wrt

#x# , we have:

# 2ay dy/dx = 3x^2 => dy/dx = (3x^2)/(2ay) # The gradient of the tangent to a curve at any particular point is given by the derivative of the curve at that point. So, at a generic point

#P(at^2,at^3)# , the derivative is:

# m_T = (3x^2)/(2ay) = (3(at^2)^2)/(2a(at^3)) = (3a^2t^4)/(2a^2t^3) = 3/2t# So the tangent passes through

#P(at^2,at^3)# and has gradient#m_T=3/2t# , and using the point/slope form#y-y_1=m(x-x_1)# the tangent equation we seek is;

# y - at^3 = 3/2t( x - at^2 ) #

# :. y - at^3 = 3/2tx - 3/2at^3 #

# :. y = 3/2tx - 1/2at^3 # Which is the sought equation.

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The equation of the tangent to the curve ay^2=x^3 at the point (at^2,at^3) is y = 2at^3/x.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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